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In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standardorthogonal complex structures on flat tori $T^{2n}_{\mathbb R}$ for any $n\geq3$. We will call these examples BSV-tori. In this note, we show that on a flat$6$-torus, all the orthogonal complex structures are either the complex tori orthe BSV-tori. This solves the classification problem for compact Hermitianmanifolds with flat Riemannian connection in the case of complex dimensionthree.

The set of all orthogonal complex structures on the flat $6$-tori